Retardation plate

How a wave plate works (λ / 2 plate)

A retardation or wave plate (also: λ / n plate ) is an optical component that can change the polarization and phase of electromagnetic waves (mostly light ) passing through . For this purpose, use is made of the fact that light propagates in birefringent, suitably oriented material, depending on the position of the polarization plane, with different wavelengths. The following types are common in crystal optics :

• A λ / 4 plate delays light that is polarized parallel to a component-specific axis by a quarter wavelength - or  π / 2 - compared to light polarized perpendicular to it. With correct irradiation, it can turn linearly polarized light into circular or elliptically polarized light and linearly polarized light from circularly polarized light and elliptically polarized light.
• A λ / 2 plate delays light that is polarized parallel to a component-specific axis by half a wavelength - or  π - compared to light polarized perpendicular to it. It can rotate the polarization direction of linearly polarized light by a selectable angle. In the case of circularly polarized light, a λ / 2 plate reverses the helicity (left or right circular polarization).

The changes in polarization come about because the light can be split into two perpendicular polarization directions which pass the retardation plate at different speeds and whose phases are shifted from one another.

Such a plate typically consists of a birefringent crystal (e.g. mica ) with a suitably selected thickness and orientation. There are also retardation plates in which a mechanically pre-stressed plastic film is cemented between two glass plates.

functionality

This figure shows the so-called ellipsoid of the refractive index for an optically positive uniaxial material with the two different refractive indices . It is also shown how a path (i.e. also phase) difference builds up between ordinary and extraordinary rays (differ in their polarization) when the crystal-optical axis is perpendicular to the direction of incidence. The different polarizations of the light are indicated above the crystal interface. The circles each indicate the position of a wavefront that moves with c / n through the crystal and originates from a single excitation center

A retardation plate is a thin disc of optically anisotropic material, that is, material which has different propagation speeds c / n (or different refractive indices n ) in different directions for differently polarized light . Often used materials are optically uniaxial, that is, there are two main refractive axes perpendicular to each other in the crystal, along which the refractive indices differ. These are called the ordinary (the E-vector of light is polarized perpendicular to the crystal-optical axis ) and extraordinary axis (the E-vector of light is polarized parallel to the crystal-optical axis). The direction of oscillation of the light, in which a wave has the greater speed of propagation, is called the “fast axis”, the direction perpendicular to it is called the “slow axis”. For retardation plates, the crystals are cut so that their crystal-optical axis lies in the plane of the polished entrance surface. The fast axis is usually marked on commercially available plates so that the alignment can be precisely determined.

This figure shows how a beam polarized at the angle α to the crystal-optical axis falls on the plate and the electric field is projected onto the fast and slow axes. Here only the crystal-optical axis (parallel to the slow axis for positively birefringent materials) is drawn.

The function of such a wave plate made of an optically positive uniaxial material (e.g. quartz ) will be described below. The slow axis coincides with the crystal-optical axis of the crystal (axis of high symmetry in the crystal lattice). The refractive indices along these axes are denoted by and . ${\displaystyle n_{\mathrm {schnell} }}$ ${\displaystyle n_{\mathrm {langsam} }}$

Light polarized parallel to the fast axis takes less time to travel through the plate than light polarized perpendicular to it. One can imagine the light divided into two linearly polarized components perpendicular (ordinary ray) and parallel (extraordinary ray) to the crystal-optical axis. After passing through the plate, the two waves show a phase shift to one another:

${\displaystyle \Delta \varphi ={\frac {2\pi }{\lambda _{0}}}\cdot d\cdot (n_{\mathrm {langsam} }-n_{\mathrm {schnell} })}$

Here d is the thickness of the plate and the vacuum wavelength of the incident light. The two waves overlap behind the crystal ( interference ) to form the outgoing light. The (coherent) superposition of these two waves results in a new polarization of the light (frequency and wavelength are retained; see next section). As can be seen in the equation, the thickness of a retardation plate has a decisive influence on the type of superposition. For this reason, such a retardation plate is only ever designed for a specific wavelength. ${\displaystyle \lambda _{0}}$

It should also be noted that splitting into two rays is just a kind of computational trick. In reality, of course, these two rays overlap at every point on the crystal. The electrons around the crystal atoms form local and momentary dipoles, which oscillate in a superposition of the two polarization directions of the beams.

λ / 4 plate

λ / 4 plate as a circular polarizer

If one chooses d in the above formula so that there is a phase shift of π / 2, one obtains a λ / 4 plate.

Animation of how a λ / 4 plate works

How a λ / 4 plate works

If a linearly polarized light beam whose direction of polarization is rotated by 45 ° to the crystal-optical axis hits the plate, circularly polarized light is produced. If the setting is different from 45 °, then generally elliptically polarized light is produced. The reason for this is that the light beam is split into two parts polarized perpendicular to one another, which overlap again at the exit of the plate, shifted by a quarter phase. This creates a Lissajous figure (circle or ellipse) for the resulting field vector of the exiting light beam , which causes a complete rotation of the plane of polarization by 360 ° during each oscillation cycle. A λ / 4 plate is therefore also called a circular polarizer . Conversely, a λ / 4 plate also converts circularly polarized light into linearly polarized light.

If, on the other hand, the direction of polarization of the incident light is parallel to one of the axes, then linearly polarized but phase-shifted light is obtained after the platelet.

Two λ / 4 plates connected one behind the other result in a λ / 2 plate if their optical axes are aligned in parallel.

λ / 2 plate

If there is a shift by π at the top, a λ / 2 plate is obtained. Such a plate can be used to rotate the plane of polarization of linearly polarized light. If the plane of polarization has the angle α to a crystal-optical axis when it enters, it has the angle −α after passing through the plate, i.e. it is rotated by the angle 2α.

Mathematical description

Consider a plane wave polarized linearly in the y direction in the z direction

${\displaystyle {\vec {E}}(z,t)={\vec {E}}_{0}\cdot \exp \left[i(\omega t-kz)\right]=E_{0}\cdot {\begin{pmatrix}0\\1\end{pmatrix}}\cdot \exp \left[i(\omega t-kz)\right].}$

The physical quantity is described by the real part of this complex quantity, i.e.:

${\displaystyle {\vec {E}}(z,t)={\vec {E}}_{0}\cdot \cos \left[(\omega t-kz)\right]}$

The vector is a vector in the x - y plane. This now hits perpendicularly on a retardation plate, the slow axis of which is tilted at the angle α to the y direction (see drawing above). We now switch to the coordinate system of the axes of the retardation plate. Then it is projected onto the axes and you get: ${\displaystyle {\vec {E}}_{0}}$ ${\displaystyle {\vec {E}}(z,t)}$

${\displaystyle {\vec {E}}(z,t)={\begin{pmatrix}E_{\|}\\E_{\bot }\end{pmatrix}}\cdot \exp \left[i(\omega t-kz)\right]=E_{0}\cdot {\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}}\cdot \exp \left[i(\omega t-kz)\right].}$

The wave plate now causes a phase delay of the slow axis (part ) compared to the fast axis, so you get: ${\displaystyle \Delta \varphi }$ ${\displaystyle E_{\bot }}$

${\displaystyle {\vec {E}}(z,t)=E_{0}\cdot {\begin{pmatrix}\cos \alpha \\e^{i\Delta \varphi }\cdot \sin \alpha \end{pmatrix}}\cdot \exp \left[i(\omega t-kz)\right].}$

For a λ / 4 plate the following applies . If one considers the real part of the complex quantity (the physical E field), the result is: ${\displaystyle e^{i\Delta \varphi }=i}$

${\displaystyle {\vec {E}}(z,t)=E_{0}\cdot {\begin{pmatrix}\cos \alpha \cdot \cos \left[(\omega t-kz)\right]\\-\sin \alpha \cdot \sin \left[(\omega t-kz)\right]\end{pmatrix}}.}$

However, this corresponds to a movement of the E field vector in the x - y plane in space and time. For α = 45 ° , a circular path is obtained for the tip of the E field vector. An ellipse results for other angles. ${\displaystyle \sin \alpha =\cos \alpha ={\frac {1}{\sqrt {2}}}}$

For a λ / 2 plate the following applies and accordingly: ${\displaystyle e^{i\Delta \varphi }=-1}$

${\displaystyle {\vec {E}}(z,t)=E_{0}\cdot {\begin{pmatrix}\cos \alpha \\-\sin \alpha \end{pmatrix}}\cdot \exp \left[i(\omega t-kz)\right].}$

This corresponds to a rotation of the polarization by the angle 2α.

These calculations can be carried out more elegantly using the Jones or Müller formalism . These are particularly suitable for combining several retardation plates or with other optical elements.

literature

• Wolfgang Demtröder: Experimental Physics 2 . Springer, 2004.
• Dieter Meschede: Gerthsen Physics . 22nd edition. Springer, 2004 ISBN 3-540-02622-3 .
• BEA Saleh, MC Teich: Fundamentals of Photonics . John Wiley, 1991.

Web links

• Script with a section on polarization optics (PDF file; 680 kB)

Individual evidence

1. ^ Low, Heinz., Eichler, Hans-Joachim., Bergmann, Ludwig., Schaefer, Clemens .: Optics . Ed .: Heinz Niederig . 9th edition De Gruyter, Berlin 1993, ISBN 3-11-012973-6 , p. 586 .